Notes: `((100),(2))` Find the binomial coefficient.

Sunday, February 21, 2016

$\displaystyle{\left(\matrix{{100}\\{2}}\right)}$ Find the binomial coefficient.

You need to evaluate the binomial coefficient, using the next formula, such that:

$\displaystyle{\left(\matrix{{n}\\{k}}\right)}=\frac{{{n}!}}{{{k}!{\left({n}-{k}\right)}!}}$

Replacing 100 for n and 2 for k yields:

$\displaystyle{\left(\matrix{{100}\\{2}}\right)}=\frac{{{100}!}}{{{2}!{\left({100}-{2}\right)}!}}\Rightarrow{\left(\matrix{{100}\\{2}}\right)}=\frac{{{100}!}}{{{2}!{\left({98}\right)}!}}$

Notice that $\displaystyle{100}!={98}!\cdot{99}\cdot{100}$

$\displaystyle{\left(\matrix{{100}\\{2}}\right)}=\frac{{{98}!\cdot{99}\cdot{100}}}{{{2}!{\left({98}\right)}!}}\Rightarrow{\left(\matrix{{100}\\{2}}\right)}=\frac{{{99}\cdot{100}}}{{{1}\cdot{2}}}={4950}$

Hence, evaluating the binomial coefficient yields $\displaystyle{\left(\matrix{{100}\\{2}}\right)}={4950}.$

Notes

Sunday, February 21, 2016

$\displaystyle{\left(\matrix{{100}\\{2}}\right)}$ Find the binomial coefficient.

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